Remove appendix
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paper.tex
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paper.tex
@ -262,9 +262,9 @@ the time dispensing water, is \cite[Section 14.3]{stewart_probability_2009}%
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,%
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\end{align*}%
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where $S$ denotes the service time (i.e., the time spent refilling a bottle),
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$\lambda$ the mean arrival time, and $\rho$ the system utilization. Using our
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experimental data we can approximate all parameters and calculate
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\todo{$W \approx 123$} (See the appendix for a complete derivation).
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$\lambda$ the mean arrival time, and $\rho = \lambda \cdot E\mleft\{ S \mright\}$ the system utilization. Using our
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experimental data we can approximate all parameters and obtain
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\todo{$W \approx 123$}.
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% We examine the effects of the choice of hydration strategy. To
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% this end, we start by estimating the potential time savings possible by always
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% choosing the fastest strategy:%
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@ -337,69 +337,69 @@ academic performance of KIT students: always pressing the right button.
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\printbibliography
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\appendix
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Derivation of Service Time}
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\label{sec:Derivation of Service Time}
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We want to compute the response time of our queueing system, i.e.,
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\cite[Section 14.3]{stewart_probability_2009}
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\begin{align*}
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W = E\mleft\{ S \mright\} + \frac{\lambda E\mleft\{ S^2 \mright\}}{2\mleft( 1-\rho \mright)}
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.%
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\end{align*}%
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We start by modelling the service time and subsequently calculate $\lambda$
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and $\rho$.
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Let $S, V$ and $R$ be random variables denoting the service time, refill volume
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and refill rate, respectively. Assuming that $V$ and $R$ are independent, we
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have
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\begin{gather*}
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S = \frac{V}{R} \hspace{5mm} \text{and} \hspace{5mm}
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P_R(r) = \left\{\begin{array}{rl}
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P(S_\text{L}), & r = r_{S_\text{L}} \\
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1-P(S_\text{L}), & r = r_{S_\text{R}}
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\end{array}\right.
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\end{gather*}%
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\begin{align*}
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E\mleft\{ S^2 \mright\} &= E\mleft\{ V^2 \mright\}E\mleft\{ 1 / R^2 \mright\} \\
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& = E\mleft\{ V^2 \mright\} \mleft( P(S_\text{L})\frac{1}{r^2_{S_\text{L}}} + \mleft( 1 - P(S_\text{L}) \mright)\frac{1}{r^2_{S_\text{R}}} \mright)
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.%
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\end{align*}
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We now use our experimental data (having calculated $v_n, n\in [1:N]$ from the
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measured fill times and flow rates) to compute
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\begin{align*}
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\left.
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\begin{array}{r}
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E\mleft\{ V^2 \mright\} \approx \frac{1}{N+1} \sum_{n=1}^{N} v_n^2 = \todo{15}\\
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P(S_\text{L}) \approx \frac{\text{\# times $S_\text{L}$ was chosen}}{N} = \todo{123} \\
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r^2_{S_\text{L}} \approx \todo{125} \\
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r^2_{S_\text{R}} \approx \todo{250}
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\end{array}
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\right\} \Rightarrow
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\left\{
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\begin{array}{l}
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E\mleft\{ S \mright\} \approx \todo{678} \\
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E\mleft\{ S^2 \mright\} \approx \todo{123}
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\end{array}
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\right.
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.%
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\end{align*}
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$\lambda$ is the mean arrival time.
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\todo{
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\textbf{TODOs:}
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\begin{itemize}
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\item Complete text describing / obtaining $\rho$ and $\lambda$
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\item Move model derivation to method section
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\item Move calculations with model to results section
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\item Add grade gain derivation
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\item Idea: Make the whole thing 2 pages and print on A3
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\end{itemize}
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}
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% \appendix
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%
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% \section{Derivation of Service Time}
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% \label{sec:Derivation of Service Time}
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%
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%
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% We want to compute the response time of our queueing system, i.e.,
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% \cite[Section 14.3]{stewart_probability_2009}
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% \begin{align*}
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% W = E\mleft\{ S \mright\} + \frac{\lambda E\mleft\{ S^2 \mright\}}{2\mleft( 1-\rho \mright)}
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% .%
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% \end{align*}%
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% We start by modelling the service time and subsequently calculate $\lambda$
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% and $\rho$.
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%
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% Let $S, V$ and $R$ be random variables denoting the service time, refill volume
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% and refill rate, respectively. Assuming that $V$ and $R$ are independent, we
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% have
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% \begin{gather*}
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% S = \frac{V}{R} \hspace{5mm} \text{and} \hspace{5mm}
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% P_R(r) = \left\{\begin{array}{rl}
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% P(S_\text{L}), & r = r_{S_\text{L}} \\
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% 1-P(S_\text{L}), & r = r_{S_\text{R}}
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% \end{array}\right.
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% \end{gather*}%
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% \begin{align*}
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% E\mleft\{ S^2 \mright\} &= E\mleft\{ V^2 \mright\}E\mleft\{ 1 / R^2 \mright\} \\
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% & = E\mleft\{ V^2 \mright\} \mleft( P(S_\text{L})\frac{1}{r^2_{S_\text{L}}} + \mleft( 1 - P(S_\text{L}) \mright)\frac{1}{r^2_{S_\text{R}}} \mright)
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% .%
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% \end{align*}
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% We now use our experimental data (having calculated $v_n, n\in [1:N]$ from the
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% measured fill times and flow rates) to compute
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% \begin{align*}
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% \left.
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% \begin{array}{r}
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% E\mleft\{ V^2 \mright\} \approx \frac{1}{N+1} \sum_{n=1}^{N} v_n^2 = \todo{15}\\
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% P(S_\text{L}) \approx \frac{\text{\# times $S_\text{L}$ was chosen}}{N} = \todo{123} \\
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% r^2_{S_\text{L}} \approx \todo{125} \\
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% r^2_{S_\text{R}} \approx \todo{250}
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% \end{array}
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% \right\} \Rightarrow
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% \left\{
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% \begin{array}{l}
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% E\mleft\{ S \mright\} \approx \todo{678} \\
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% E\mleft\{ S^2 \mright\} \approx \todo{123}
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% \end{array}
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% \right.
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% .%
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% \end{align*}
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%
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% $\lambda$ is the mean arrival time.
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%
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% \todo{
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% \textbf{TODOs:}
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% \begin{itemize}
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% \item Complete text describing / obtaining $\rho$ and $\lambda$
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% \item Move model derivation to method section
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% \item Move calculations with model to results section
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% \item Add grade gain derivation
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% \item Idea: Make the whole thing 2 pages and print on A3
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% \end{itemize}
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% }
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\end{document}
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